Expected Wealth Under Independent Symmetric Daily Stock Returns
Summary
This note considers a stock whose daily return is equally likely to be a gain or loss of the same percentage, with no dividends or interest. It asks whether buying or selling has an expected-value advantage under those assumptions. Starting from one unit of capital, it models each day's price change as an independent multiplier and shows that the expected multiplier is one. Independence then makes expected wealth after multiple days equal to its starting value.
The explanation addresses why an up move followed by a down move does not by itself determine the expected result: all possible paths must be considered, including consecutive moves in the same direction. The conclusion concerns expected wealth, not the desirability of holding the stock for an investor with risk preferences or other company-related considerations. It also illustrates that changing the down-state multiplier to the reciprocal of the up-state multiplier changes the expected multiplier, so the result depends on the precise return model. No market data or empirical test is presented.
Key ideas
- With independent daily returns that are equally likely to be positive or negative by the same percentage, the expected daily wealth multiplier is one.
- Independence means expected wealth remains at its initial level over multiple days under this model.
- An up move followed by a down move is only one possible path and does not establish the overall expected outcome.
- Expected wealth does not account for an investor's attitude toward risk.
- Using reciprocal price multipliers instead of symmetric percentage changes can produce a different expected return.
Tags
Full text
# The right choice when the price of a stock follows a random walk
# The right choice when the price of a stock follows a random walk
I've got the following question:
> Suppose the price of a stock either rises or falls by the same percentage for each day. Suppose there is no dividend and the interest rate is 0. Should I buy the stock now or sell it? Is there no difference?
I think there were more conditions to this problem but I cannot remember. Owning the stock gives some rights of the company, so I think we should also assume that the prospect of the company is neutral as well... I couldn't understand the intention and the answer of the problem. I would really appreciate if someone can explain it.
## Answer by Bjørn Kjos-Hanssen (score 3, accepted)
https://quant.stackexchange.com/a/36692
It makes no difference. Starting with a capital of 1, let $X_i$ be the multiplying factor for the $i$th day, so $X_i\in\{1+r,1-r\}$ with each possibility having probability 1/2. The expected capital after one day is $$\mathbb E(X_1)=\frac12((1+r)+(1-r))=1.$$ After $n$ days, your capital is $X_1X_2\cdots X_n$, and $$\mathbb E(X_1\cdots X_n)=\mathbb E(X_1)\cdots\mathbb E(X_n)=1$$ since the days are independent.
#### Clarifying notes
- You might think,
> "$(1+r)(1-r)=1-r^2<1$ so as the stock goes up and down, I lose money"
but note that if the stock goes down, then down again, you have $$(1-r)^2=1-2r+r^2>1-2r,$$ so you get more than what you'd get using a "simple interest" idea, and this together with the up/up case cancels out the loss in the up/down case.
Indeed, for $n=2$ the expected capital is $$\frac14[(1+r)^2+2(1+r)(1-r)+(1-r)^2)]=1.$$
- Of course if you like or dislike risk then you may want to buy or not buy, respectively, but the expected capital remains 1.
- On the other hand if the multipliers were not $\{1+r,1-r\}$ but $\{1+r, \frac1{1+r}\}$ then it would make sense to buy, as the expectation after one day would be $$\frac12\left(1+r+\frac1{1+r}\right)=\frac12\frac{(1+r)^2+1}{1+r}=1+\frac{r^2}{2(1+r)}>1.$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.